Calculus for Economics, Exercises (Dec. 04,2017)

I

Let $p,q,I>0$. We try to maximize the utility function $$ u(x,y)=x^{\frac 13}y^{\frac 13} $$ under the constraint $$ I-px-qy=0 $$

(1) Find the statinary point of $f(x,y$ subject to $g(x,y=0$ to find $$ x(p,q,I)=\frac I{2p},\quad y(p,q,I)=\frac I{2q},\quad \lambda=\frac 13\sqrt[3]{\frac 2{pqI}} $$

(2) We define the indirect utility function by $$ v(p,q,I)=u(x(p,q,I),y(p,q,I)) $$ and show $$ \frac{\partial v}{\partial I}=\lambda(p,q,I) $$

(3) Show that the Roy's identies $$ \frac{\partial v}{\partial x}+ \frac{\partial v}{\partial I}\cdot x(p,q,I)=0 $$ hold.

II

Let $p,q,I>0$. We try to maximize the utility function $$ u(x,y)=\frac 13\log x+\frac 13\log y $$ subject to the constraint $$ I-px-qy=0 $$

Find the demand functions $x(p,q,I)$ and $y(p,q,I)$ and the marginal utility function of the budget $\lambda(p,q,I)$.

III

Optimize $$z=x+y$$ under the constraint $$ x^2+2y^2-24=0 $$

IV

Let $p,q,I>0$. We try to maximize the utility function $$ u(x,y)=x^{\frac 13}y^{\frac 23} $$ under the constraint $$ I-px-qy=0 $$

Find the demand functions $x(p,q,I)$ and $y(p,q,I)$ and the marginal utility function of the budget $\lambda(p,q,I)$.

V

We consider $z=f(x,y)=x^2y$ subject to the constraint $g(x,y):=2x^2+y^2-1=0$.

VI

A function in $x$, $y=\varphi(x)$ satisfies the identity \begin{equation*} x^2+\varphi(x)^2-3x\varphi(x)=0 \end{equation*} Express $\varphi'(x)$ and $\varphi''(x)$ by $x$ and $\varphi(x)$.